According to “Megadeath” at Discover.com it was a hemorraghic fever, native to Mexico and only found when severe drought is followed by a wet period. The Aztecs knew smallpox; they even had their own name for it. But the horrible death that wiped most of them out, they called something else.
It is fascinating reading. (If you don’t have time to read the whole thing, Comment 2 has relevant short quotes from the article.)
Thanks to Mirabilis for the pointer.